Basically does it even matter what the reputation is when it comes to adding it to the Circle tests, as long as you’re in the area where you actually have said reputation.
For instance does it matter if I have a 1D reputation as a “Fastest dude in the town” or 1D reputation for “Saving a mayor’s son from being run over by a horse” I will still add that 1D to whatever Circles roll I do in the same town?
My answer to your question there is, “No, but actually yes.”
No, because you have the 1D Reputation, which gives you an extra die when circling in the Rep’s purview. Clear-cut, easy-peasy.
Yes, because context is important. If you have a Rep for having saved the mayor’s son, you could probably squeeze another Advantage die or two from me when you are trying to Circle up the Mayor himself. (At least, for the first time or two. I’m a busy guy; what have you done for me lately?) On the hand, if I know the “tragic accident” that almost killed the mayor’s son was actually an assassin’s plot, and you try to Circle up an assassin, now you have an Infamous Reputation for this test because you’ve thwarted them before! The enmity clause features a desire for payback, an you’d wish you were the fastest dude in town! What might have been a simple failure or a requirement for a little quid-pro-quo (“Punch that guard and then run away so I can slip through the gate. You’re the fastest guy in town, right? You’ll get away easy.”) becomes a life-or-death Fight!
Reputations can also be factored into Relationship Aptitudes. You have an easier time forming a Relationship with the mayor if you’ve saved his son. As the fastest dude in town, you might build Relationships with craftsmen or shop-keepers (people who lean on couriers a lot).
In my estimation, there isn’t really a whole lot of “fluff” in Burning Wheel. The mechanics are so sensitive to context that just about anything in the fiction can be leveraged for or against a character.
If I were GMing, I’d probably share @Gnosego’s generosity.
Context also goes the other way: if you have a reputation in a village for something then people will gossip about it, which means travellers will hear about it. So, I might allow a Reputation to benefit Circles tests over a wider area if it was extremely relevant. For example, if you have 2D “best archer in the village”, then you get 2D to Circles in the village and 1D to Circles in the nearby town when archery related because some stories have reached people who take note of archers.
That reminds me - your Reputation can be used as Advantage dice when people are trying to to Circle up you. This is doesn’t really come up for players, I find (if the GM needs an NPC to find a player for the sake of the story, they usually just do.), but still, it can help color and inform those NPCs and interactions.
Which triggers in turn a thought in me: celebrity makes it harder to not be noticed, so a player’s reputation could give advantage dice to an NPC in a Versus test to not be found. For example, if the player spots a man they want to avoid coming into a crowded pub and wants to fade into the background, it will be a lot easier for their enemy to spot them if they have a 3D Reputation for “playing the lute really well” because of all the people clustered around begging them to play the lute.
A 3D Reputation for “Playing the lute really well”! You’re Van Halen! … Or Dandelion? 
Anyway. That sounds like an Ob Penalty to Inconspicuous to me!
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